A fast simulation method for 1D heat conduction

A flexible solution method for the initial-boundary value problem of the temperature field in a one-dimensional domain of a solid with significantly nonlinear material parameters and radiation boundary conditions is proposed. A transformation of the temperature values allows to isolate the nonlinear material characteristics into a single coefficient of the heat conduction equation. The Galerkin method is utilized for spatial discretization of the problem and integration of the time domain is done by constraining the boundary heat fluxes to piecewise linear, discontinuous signals. The radiative heat exchange is computed with the help of the Stefan-Boltzmann law, such that the ambient temperatures serve as system inputs. The feasibility and accuracy of the proposed method are demonstrated by means of an example of heat treatment of a steel slab, where numerical results are compared to the finite difference method.